# New high precision orbital and physical parameters of the double-lined low-mass spectroscopic binary BY Draconis

###### Abstract

We present the most precise to date orbital and physical parameters of the well known short period ( d), eccentric () double-lined spectroscopic binary BY Draconis, a prototype of a class of late-type, active, spotted flare stars. We calculate the full spectroscopic/astrometric orbital solution by combining our precise radial velocities (RVs) and the archival astrometric measurements from the Palomar Testbed Interferometer (PTI). The RVs were derived based on the high resolution echelle spectra taken between 2004 and 2008 with the Keck I/HIRES, Shane/CAT/HamSpec and TNG/SARG telescopes/spectrographs using our novel iodine-cell technique for double-lined binary stars. The RVs and available PTI astrometric data spanning over 8 years allow us to reach 0.2-0.5% level of precision in and the parallax but the geometry of the orbit () hampers the absolute mass precision to 3.3%, which is still an order of magnitude better than for previous studies. We compare our results with a set of Yonsei-Yale theoretical stellar isochrones and conclude that BY Dra is probably a main sequence system more metal-rich than the Sun. Using the orbital inclination and the available rotational velocities of the components, we also conclude that the rotational axes of the components are likely misaligned with the orbital angular momentum. Given BY Dra’s main sequence status, late spectral type and the relatively short orbital period, its high orbital eccentricity and probable spin-orbit misalignment are not in agreement with the tidal theory. This disagreement may possibly be explained by smaller rotational velocities of the components and the presence of a substellar mass companion to BY Dra AB.

###### keywords:

binaries: spectroscopic – binaries: visual – stars: fundamental parameters – stars: individual (BY Dra) – techniques: interferometric – techniques: radial velocities^{†}

^{†}pagerange: New high precision orbital and physical parameters of the double-lined low-mass spectroscopic binary BY Draconis–References

^{†}

^{†}pubyear: 2011

## 1 Introduction

Regular studies of BY Dra (Gl 719, HD 234677, HIP 91009, NLTT 46684, BD+51 2402) started in mid 40’s when Münch noted the calcium H and K lines to be in emission (Münch, 1944). This fact was later confirmed by Popper (1953) who also noted strong emission Balmer series lines. In one of his spectrograms the emission was particularly strong which led to a conclusion that BY Dra may be a member of a new group of flare stars (Popper, 1953). Photometric monitoring was then carried out (e.g. Masani et al., 1955) but the variability was not confirmed until 1966 when Chugainov obtained a quasi-sinusoidal light curve with an amplitude of 0.23 mag and a period of 3.826 d (Chugainov, 1966), later interpreted as a rotation of a spotted star (Krzemiński, 1969). No flares were then observed. The first photometric flares were reported by Cristaldi & Rodono (1968) who later observed twelve flares that occurred between July 1967 and July 1970 (Cristaldi & Rodono, 1970, 1971). Krzemiński (1969) confirmed the sinusoidal variability with a 3.826 d period and noted the variation of its amplitude. Many subsequent studies of BY Dra’s variability have been carried out and the most up-to-date value of the rotational period d is given by Pettersen, Olah & Sandmann (1992) as an average period for their entire 1965-1989 data set.

The double-lined spectroscopic nature was revealed by Krzemiński & Kraft (1967). They announced a period of 5.981 d but never published their full orbital solution. It was done later by Bopp & Evans (1973) on the basis of 23 spectra taken between June 1966 and July 1971 at the Hale and McDonald observatories, 15 of which showed unblended Ca II H and K lines. Bopp & Evans (1973) also performed an analysis of the spots on the surface of BY Dra and estimated the rotational velocity of the primary (spotted) component to be 5 km s and the (rotational) inclination to be 30 deg. Since then BY Dra became a prototype of a new class of stars characterized by a late type, brightness variation caused by spots, rapid rotation and strong emission in H and K lines. The short orbital period also seems to be a characteristic for most of BY Dra-type stars (Bopp, Noah & Klimke, 1980).

The orbital solution was later improved by Vogt & Fekel (1979) on the basis of high-resolution reticon spectra. Vogt & Fekel (1979) also found the projected rotational velocity of the primary to be 8.5 km s under the assumption of the rotational inclination being the same as the orbital one (spin-orbit alignment). They estimated the radius of the primary to be greater than 0.9 R which led to a conclusion that BY Dra was a pre-main-sequence system. This conclusion was supported by the large brightness ratio despite the mass ratio being close to 1 (), the Barnes-Evans visual surface brightness relation (Barnes, Evans & Moffet, 1978) and the inequality of the rotational and orbital periods. However, the assumption of the spin-orbit alignment in close binary systems was later criticized in several works, e.g. Głȩbocki & Stawikowski (1995). The orbital parameters as well as the value of the projected rotational velocities for both components were shortly after improved by Lucke & Mayor (1980). They used new measurements from CORAVEL and obtained rotational velocities and km s and the mass ratio = 0.89 significantly more different from 1 than that of Vogt & Fekel (1979). They also estimated the magnitude difference (1.15 mag) and the primary’s radius (1.2 - 1.4 R) but noted that a higher macroturbulence velocity would reduce the radius estimation by a factor of 2.

The most up-to-date spectroscopic orbital solution was given together with the first astrometric solution by Boden & Lane (2001). They combined the archival RV measurements with the visibility based () astrometric measurements obtained with the Palomar Testbed Interferometer (PTI; Colavita et al., 1999) in 1999. Their orbital inclination (152 deg, retrograde orbit) agrees with the first estimations of the rotational inclination (Bopp & Evans, 1973) but not with the later ones (Głȩbocki & Stawikowski, 1995).

Finally it is worth noting that BY Dra is a hierarchical multiple system. A common proper-motion companion was found by Zuckerman et al. (1997) about 16.7 arc-sec to the northeast of the primary. From the visual and infrared photometry of BY Dra C they also deduced that this component is a normal M5 dwarf at least yr old which makes the pre-main-sequence nature of BY Dra less probable. Yet another putative component is reported in the Hipparcos Double and Multiple System Annex (ESA, 1997). A photocentric circular orbital solution with a period of 114 d and 113 deg inclination is reported. Boden & Lane (2001) however demonstrated that this is an improbable solution since the 4-th body would produce significant perturbations to the BY Dra AB radial velocities but no such periodicity is seen in the archival RVs.

We spectroscopically observed BY Dra over the years 2004-2008 using a combination of high resolution echelle spectrographs HIRES (10-m Keck I), SARG (3.5-m TNG) and HamSpec (3-m Shane telescope) as a part of our ongoing RV search for circumbinary planets (Konacki et al., 2009, 2010). Even though we knew that BY Dra was too variable to allow us reach an RV precision sufficient to detect planets, it was nevertheless observed to make use of an extensive and publicly available set of PTI measurements spanning now over 8 years.

In this paper we present a new orbital solution and the orbital and physical parameters of the BY Dra AB binary, derived with a precision of over an order of magnitude better than by Boden & Lane (2001). Thanks to our superior iodine-cell-based radial velocities and the full set of PTI visibilities we are able to put strong constraints on the nature of the system. In Sections 2 and 3 we present the data — s and RVs. In Section 4 we describe their modeling. The results of our data modeling are presented in Section 5 and the state of BY Dra is then discussed in Section 6.

## 2 Visibilities

TDB - 2400000 | Tel./Spec. | ||||||||
---|---|---|---|---|---|---|---|---|---|

(km s) | (km s) | (km s) | (km s) | (km s) | (km s) | (km s) | (km s) | ||

53276.214636 | -49.47297 | 0.00668 | 0.15015 | 0.05236 | 1.74203 | 0.01162 | 0.15045 | -0.04117 | K/H |

53276.218251 | -49.29837 | 0.00822 | 0.15022 | 0.06779 | 1.57802 | 0.01371 | 0.15063 | -0.02414 | K/H |

53276.274388 | -46.62269 | 0.00994 | 0.15033 | 0.14884 | -1.24054 | 0.01193 | 0.15047 | 0.10849 | K/H |

53276.277251 | -46.46912 | 0.01018 | 0.15034 | 0.16444 | -1.39294 | 0.01152 | 0.15044 | 0.11303 | K/H |

53276.371688 | -41.65463 | 0.00990 | 0.15033 | 0.19386 | -6.80759 | 0.01758 | 0.15103 | 0.14139 | K/H |

53276.381066 | -41.16056 | 0.00934 | 0.15029 | 0.19369 | -7.39337 | 0.02663 | 0.15235 | 0.11782 | K/H |

53276.383829 | -41.01684 | 0.01020 | 0.15035 | 0.19138 | -7.57152 | 0.02920 | 0.15282 | 0.10579 | K/H |

53328.260862 | -34.23894 | 0.01017 | 0.15034 | -0.13033 | -15.80064 | 0.01163 | 0.15045 | -0.00187 | K/H |

53329.192929 | -55.69433 | 0.00958 | 0.15031 | -0.36569 | 8.16117 | 0.01185 | 0.15047 | -0.19837 | K/H |

53567.417144 | -37.51257 | 0.00765 | 0.15019 | -0.07442 | -12.24447 | 0.01409 | 0.15066 | -0.23557 | K/H |

53654.287570 | -2.81022 | 0.00707 | 0.15017 | -0.26482 | -51.63684 | 0.01257 | 0.15053 | 0.04275 | K/H |

53655.270877 | -8.62844 | 0.00867 | 0.15025 | -0.20946 | -44.96407 | 0.00776 | 0.15020 | 0.05011 | K/H |

53656.250010 | -21.53447 | 0.01434 | 0.15068 | 0.29488 | -29.89713 | 0.02974 | 0.15292 | -0.12965 | K/H |

54191.188978 | -13.36607 | 0.02168 | 0.15156 | 0.10101 | -39.92790 | 0.03336 | 0.15366 | -0.10395 | T/S |

54192.164756 | -2.95488 | 0.01336 | 0.15059 | -0.10836 | -52.02801 | 0.03054 | 0.15308 | -0.10339 | T/S |

54247.147226 | -12.62161 | 0.02067 | 0.15142 | -0.03078 | -40.66218 | 0.02748 | 0.15250 | 0.19585 | T/S |

54275.086679 | -7.54149 | 0.01298 | 0.15056 | 0.03906 | -46.51656 | 0.02529 | 0.15212 | 0.00909 | T/S |

54281.357833 | -3.42439 | 0.02136 | 0.15151 | 0.18914 | -50.40787 | 0.02344 | 0.15182 | 0.02977 | S/H |

54290.431083 | -38.18861 | 0.00594 | 0.15012 | -0.18164 | -11.42422 | 0.00905 | 0.15027 | -0.06282 | K/H |

54290.596520 | -41.90239 | 0.00805 | 0.15022 | -0.07812 | -7.00027 | 0.01280 | 0.15055 | 0.01660 | K/H |

54727.249888 | -52.87236 | 0.01649 | 0.15090 | -0.09239 | 5.11252 | 0.02114 | 0.15148 | -0.36239 | S/H |

54728.248858 | -45.72092 | 0.02849 | 0.15268 | -0.09889 | -2.56209 | 0.02087 | 0.15145 | 0.07559 | S/H |

54752.198503 | -43.21949 | 0.01716 | 0.15098 | -0.08641 | -5.07445 | 0.01632 | 0.15089 | 0.39454 | S/H |

54789.116481 | -4.71313 | 0.02199 | 0.15160 | 0.08852 | -49.23474 | 0.04418 | 0.15637 | -0.14975 | S/H |

Often the main observable in the interferometric observations at optical or infrared wavelength is the normalized amplitude of the coherence function – a fringe pattern contrast, commonly known as the visibility (squared, V) of the interferometric fringes, calculated by definition as follows (Boden, 1999):

(1) |

where and are the maximum and minimum intensity of the fringe pattern respectively. For a given object the observed depends on its morphology and the projected baseline vector of a two-aperture interferometer onto a plane tangent to the sky. For binaries, varies also due to the orbital motion of the components. In the case of a binary, approximated by two uniform disks, the squared visibility can be modeled as follows (see e.g. Boden, 1999):

(2) |

where are the visibilities of uniform disks (components) of the angular diameters and and are calculated as follows:

(3) |

where is the brightness ratio at the observing wavelength , is the first order Bessel function and is the separation vector between the primary and the secondary in the plane tangent to the sky. This vector is related to the Keplerian orbital elements, orbital period , eccentric anomaly (from the Kepler equation ), and the parallax in the usual way (van de Kamp, 1967).

A visibility measurement needs to be calibrated by observing at least one calibration source before or after a target observation. The calibrator is typically a single star with a known diameter and its visibility is given by Relation 3. The correction factor which should be applied to the observed target is simply the ratio where is the measured calibrator visibility. The “true” target visibility is then

(4) |

Uncalibrated visibilities of BY Dra were extracted from the NASA
Exoplanet Science Institute (NExSci) database of the PTI
measurements^{1}^{1}1https://nexsciweb.ipac.caltech.edu/pti-archive/secure/main.jsp.
These measurements were made in (2.2 m) and (1.6 m)
bands. They were calibrated using the standard tools provided by NExSci
(getCal and wbCalib). As the calibration objects we used
HD 177196 (A7V, mag, mag, diameter mas,
6.6 deg from BY Dra) and HD 185395
(F4V, mag, mag, mas, 9.9 deg) as in Boden & Lane (2001).
We do not list these measurements as they can be easily obtained using
the NExSci database and tools.

## 3 Radial velocities

Our high-resolution echelle spectra of BY Dra were obtained during 17
nights between September 2004 and November 2008. We collected 24 spectra
using Keck I/HIRES (K/H, 15 spectra), TNG/SARG (T/S, 4) and
Shane/HamSpec (S/H, 5) telescopes/spectrographs. Our spectra have the
resolutions R 67 000 for K/H, 86 000 for T/S and 60 000 for S/H.
The typical signal to noise ratio () per collapsed pixel at 550 nm was
250 for K/H, 90 for T/S and 60 for S/H.
The basic reduction (bias, dark, flatfield, scattered light subtraction) was
done with the ccdred and echelle packages from
iraf^{2}^{2}2iraf is written and supported by the
iraf programming group at the National Optical Astronomy
Observatories (NOAO) in Tucson, AZ. NOAO is operated by the Association of
Universities for Research in Astronomy (AURA), Inc. under cooperative
agreement with the National Science Foundation. http://iraf.noao.edu/.
The wavelength solution and radial velocities were obtained with our
novel procedure based on the iodine cell technique (Konacki, 2009; Konacki et al., 2009, 2010).
This procedure employs a tomographic disentangling of the component
spectra of double-lined spectroscopic binaries (SB2s) implemented
through a maximum entropy method and the two-dimensional cross-correlation
technique todcor (Zucker & Mazeh, 1994) using synthetic spectra derived with
atlas 9 and atlas 12 codes (Kurucz, 1995) as templates
for the first approximation of the RVs. With this
approach it is possible to reach up to 2 m s precision in RVs
for components of SB2s (Konacki et al., 2009) but in the case of BY Dra
the precision is hampered by the activity of the star (presence of
spots) and the relatively rapid rotation of both components.

In Table 1 we list our RV measurements together with their uncertainties and the best-fit s. The formal errors, , were calculated from the scatter between orders and predominantly reflect a high SNR of our spectra. The formal errors underestimate the true RV scatter (due to activity) and the resulting reduced of the spectroscopic orbital fit was much larger than 1. Hence to obtain a conservative estimation of the parameters’ errors (and the reduced close to 1) we added in quadrature a systematic error of 150 m s. Let us note that spots can easily induce RV variations at the level of a few hundreds of m s so the RV variability of BY Dra is not surprising (see e.g. Hełminiak & Konacki, 2011; Hełminiak et al., 2011). We also had to adopt small shifts between each data set as is explained in Konacki et al. (2010). The best fit values of the shifts can be found in Table 3 in Section 5. We do not include the CORAVEL data (from Lucke & Mayor, 1980) since their precision is substantially worse than ours.

## 4 Modeling

We combined all and RV measurements in a simultaneous least-squares fit to derive the full orbital solution and the physical parameters of BY Dra. We used our own procedure which minimizes the function with a least-squares Levenberg-Marquardt algorithm. The procedure fits a Keplerian orbit with corrections to the RVs due to tidal distortions of the components and relativistic effects. In order to model the tidal term we use the Wilson-Devinney (WD) code (Wilson & Devinney, 1971) as is explained in Konacki et al. (2010) and assume several parameters of BY Dra listed in Table 2. Note that both the relativistic and tidal effects are much smaller than the RV scatter (see Fig. 1) but we decided to include them in the RV model anyway to maintain a consistent treatment of our iodine cell based RVs as in Konacki et al. (2010). Apparent stellar diameters were assumed to agree with the estimates of the radii from Section 6.2, since the components are too small and act like point sources.

For a combined +RV solution our software evaluates the the period , standard Keplerian elements: major semi-axis (of B relatively to A – apparent astrometric in mas), inclination , eccentricity , longitude of pericenter , longitude of ascending node , time of periastron passage ; velocity amplitudes and , systemic velocity , flux ratios in the observing bands and , and a set of shifts in radial velocities between the two components as well as between the data sets from each telescope/spectrograph. On this basis the software calculates such absolute physical parameters like the absolute major semi-axis (relatively to the baricentre – in AU), absolute components’ masses and , magnitude differences and , and parallax . The uncertainty of every parameter is a combination of formal best-fit least-squares errors and systematic errors as is explained in Konacki et al. (2010). For the systematic errors we assumed the following estimates for additional uncertainties related to the data reduction (1) 0.01 percent in the baseline vector coordinates, (2) 0.5 percent in and (3) 10 percent in the calibrator and binary components diameters. For the RVs we assumed (4) 10 percent in all the parameters from Table 2 except for the temperatures for which we assumed an uncertainty of 2 percent and for the metallicities we assumed an uncertainty of 0.05 dex.

## 5 Results

Parameter | Primary | Secondary |
---|---|---|

Effective temperature, (K) | 4000 | 4000 |

Potential, | 24.0 | 24.5 |

Synchronization factor, | 1.95 | 1.95 |

Gravity darkening exponent, | 0.3 | 0.3 |

Albedo, | 0.5 | 0.5 |

Apparent diameter, (mas) | 0.6 | 0.5 |

Metallicity | 0.0 |

Parameter | Value() |
---|---|

Orbital solution | |

Apparent major semi-axis, (mas) | 4.4472(91) |

Period, (d) | 5.9751130(46) |

Time of periastron, (TDB-2450000.5) | 3999.2144(21) |

Eccentricity, | 0.30014(62) |

Longitude of periastron, (deg) | 230.33(17) |

Longitude of ascending node, (deg) | 152.30(10) |

Inclination, (deg) | 154.41(29) |

Magnitude difference in K band, (mag) | 0.530(11) |

Magnitude difference in H band, (mag) | 0.60(23) |

Velocity amplitude, primary, (km s) | 28.394(60) |

Velocity amplitude, secondary, (km s) | 32.284(61) |

Mass ratio, | 0.8795(25) |

Gamma velocity, (km s) | -25.484(46) |

Velocity offsets (all in km s) | |

Secondary vs primary | -0.088(67) |

SARG vs HIRES, primary | -0.216(104) |

SARG vs HIRES, secondary | -0.343(105) |

HamSpec vs HIRES, primary | 0.076(83) |

HamSpec vs HIRES, secondary | -0.067(85) |

Least-squares fit parameters | |

Number of RV measurements, total | 48 |

Number of RV measurements, HIRES | 30 |

Number of RV measurements, SARG | 8 |

Number of RV measurements, HamSpec | 10 |

Number of V measurements | 299 |

Combined RV rms, prim./sec. (km s) | 0.169/0.157 |

Visibilities rms | 0.0312 |

RV , primary/secondary | 29.17/24.65 |

Visibilities | 395.3 |

Degrees of freedom, | 330 |

Total reduced , | 1.361 |

Parameter | Primary | Secondary |
---|---|---|

Major semi-axis, (10 AU) | 3.4437(73) | 3.9155(74) |

Major semi-axis, (R) | 7.400(16) | 8.414(16) |

sin (M) | 0.06387(28) | 0.05618(26) |

Mass, (M) | 0.792(26) | 0.697(23) |

M (mag) | 4.269(21) | 4.799(22) |

M (mag) | 4.420(86) | 5.020(149) |

Parallax, (mas) | 60.43(12) | |

Distance, (pc) | 16.548(35) |

The results of our modeling are collected in Tables 3 and 4. Figure 2 shows our RVs together with the best fitting orbital solution and the corresponding residuals and their histograms. Figure 3 shows the same for the PTI measurements. The resulting astrometric orbit of component B relative to A is shown in the panel (d). In Table 3 we show the orbital parameters for BY Dra, the velocity offsets and other parameters related to the quality of the fit. The absolute physical parameters are listed in Table 4.

As one can see, we were able to reach 0.2 % of precision in velocity amplitudes, despite such obstacles like the presence of spots or some rotational broadening of spectral lines. This level of quality has direct implication for the precision of mass ratio (0.28 %), (0.44 and 0.46 % for the primary and secondary respectively) or major semi-axis (0.2 % both for the apparent and absolute values). The level of precision in also proves that the quality of the astrometric solution is very high. The 299 visibility measurements used provide good orbital phase coverage. The apparent and physical values of major semi-axis allow us to determine the parallax, thus the distance to the system, with a precision also close to 0.2 %. Our value of the parallax – 60.43(12) mas – is in a relatively good agreement but almost 6 times more precise than 61.15(68) mas from the new reduction of the Hipparcos data (van Leeuwen, 2007). We were also able to precisely derive the magnitude difference in the band (282 measurements) but the accuracy for the band is much lower due to a lower number of measurements in (only 17).

Our final error in the absolute masses of the BY Dra components is however much higher – 3.3 % for both the primary and secondary. This is mainly due to the inclination of the orbit of 154.4 deg. For such configurations, far from edge-on, a small error in the angle propagates to a large error in the masses. Still it is a considerably more accurate measurement compared to Boden & Lane (2001) of respectively 23% and 25% for the primary and secondary. This is possible thanks to our superior RV data set (rms of 0.15 km s vs 2.3 km s) and a longer time span of the astrometric data.

## 6 Discussion

### 6.1 Age and metallicity

As it is pointed out by Torres, Andersen & Giménez (2010), the mass uncertainty should be below 3 % to be useful to perform reliable tests of the stellar evolution models. Our precision is close to that but to go below 3 % we would require a higher number of precise RVs or more measurements. Nevertheless, with our measurements we still can place some constraints on the evolutionary properties of BY Dra. We focused on the age estimation to confirm or exclude the pre-main-sequence nature of the system.

We compared our results with the Yonsei-Yale isochrones
(Y; Yi et al., 2001; Demarque et al., 2004). We used our estimations of the magnitude
differences, parallax and the apparent and magnitude from 2MASS
(Cutri et al., 2003) to derive the absolute and magnitudes of each
component separately. Using the transformation equations from
Carpenter (2001, with updates)^{3}^{3}3http://www.astro.caltech.edu/jmc/2mass/v3/transformations/
we transformed them to the ESO photometric system (van der Bliek, Manfroid & Bouchet, 1996) ,
in which the Y isochrones are available. In Figure 4
we show our measurements in the mass/-band (left) and mass/-band
(right) absolute magnitude diagrams.
For comparison we plot the isochrones for ages of
60 Myr (dashed) and 1 Gyr (solid lines) for three values of the metallicity:
= 0.02 (red), 0.04 (green) and 0.06 (blue). One can see that
the properties of the secondary component are not reproduced by the 60 Myr
isochrones, which means that it has already settled down on the main sequence.
The formally best match is found for Gyr and .
No match was found for ages below 60 Myr for any value
nor for any age value for . For Gyr only isochrones with
metallicities higher than solar reproduce the data points.
We thus conclude that BY Dra is probably
between 0.2 and 5 Gyr old and is more metal-rich than the Sun. The
most probable values of age and are 1-2 Gyr and 0.04 respectively
These facts make the pre-main-sequence scenario less probable.

One should notice that the error bars in the masses are enlarged mainly by the uncertainty in the inclination. Any change in would shift both components in the same direction – towards higher or lower masses which would definitely not improve the fit. We also have a large uncertainty in , induced by the error in , which is so large due to a small number of measurements in this band. Reduction of this uncertainty would allow for putting even more stringent constrains on the nature of the system. At the same time, is very well constrained and shows that the mass ratio is not inconsistent with the observed flux ratio, at least in the K band. Using the Y set of isochrones we can estimate that the expected theoretical magnitude difference in for the stars having 0.792 and 0.697 M, should be close to 0.9 mag. This is not in agreement with mag predicted by Lucke & Mayor (1980). However given even 0.2 mag variation from spots (Chugainov, 1966; Pettersen, Olah & Sandmann, 1992), we can conclude that such a difference in V is possible for BY Dra even if it is a main-sequence system.

To put additional constrains on the system’s age, we further
calculated the galactic space velocities ^{4}^{4}4Positive
values of , and indicate velocities toward the Galactic center,
direction of rotation and north pole respectively relatively to the
local standard of rest (LSR; Johnson & Soderblom, 1987). We applied our values of
radial systemic velocity and distance estimation together with
proper motion of mas yr and
mas yr from the PPMX catalogue (Röser et al., 2008).
Values of , and
km s put BY Dra outside of any known young moving
group or group candidate (Zhao et al., 2009), and at the transition area between the thin and
thick galactic disk (Bensby et al., 2003; Nordström et al., 2004). This supports the possibility
of BY Dra being not a PMS system.

### 6.2 Spin-orbit (mis)alignment

Using the masses and isochrones, we can estimate the radii of each component of BY Dra. In Figure 5 we plot the Y isochrone for 1 Gyr and in a radius/mass plane. As solid horizontal lines we plot the masses together with their uncertainties (dashed lines). Other probable isochrones are very close to the chosen one and do not change the results of our analysis significantly.

Our results predict values of and R. We can use this together with the rotational velocities and rotational period values from the literature to estimate the orbital inclination angles. For this purpose we use and km s from Lucke & Mayor (1980) and d from Pettersen, Olah & Sandmann (1992). This implies and . Most of the authors attribute spots to the primary component and refer the 3.8 d period to its rotation. If so, from the values above we can estimate the rotational inclination of the primary to be or deg. If we assume the secondary to rotate with the given period, we end up with or 113 deg and its uncertainties ranging from 50 to 130 deg. This means that deg is also possible. Our results are in a good agreement with values given by Głȩbocki & Stawikowski (1995) who derived and deg. None of the values however agrees with the orbital inclination deg. This indicates the spin-orbit misalignment in the BY Dra system. Even if we consider that the theoretical radii are underestimated by about 15%, a well known issue for late-type stars, we are not able to reproduce the observed .

In the case of a spin-orbit alignment, and the literature values of , the radii would have to be and R. This would occur if the system was 3-4 Myr old, depending on the metallicity. In such a case both stars should be much brighter in the infrared than it is observed. The predicted -magnitudes difference for the two stars would be 0.2 and not 0.53 mag which is observed. One also would expect the system to be a member of a young cluster, containing leftovers of the primodial gas, but this is not observed as well.

However, the spin-orbit alignment should be observed for such a close
pair of 1 Gyr old stars (Hut, 1981). The source of the
discrepancy could be for example an overestimated rotational velocity.
Głȩbocki & Stawikowski (1995) in their analysis adopted a value of 3.6 km s,
given by Strassmeier et al. (1993)^{5}^{5}5In fact, in the current version of their
catalogue, Strassmeier et al. (1993) cite values from Lucke & Mayor (1980)..
For this value of we get
deg, which is very close to the observed
orbital inclination. Hence the spin-orbit alignment
may in fact be present in the BY Dra system if Lucke & Mayor (1980) have
overestimated their rotational velocity measurements by adopting
a too small macroturbulence velocity.

Finally, let us note that a spin-orbit misalignment could manifest itself through its impact on the apsidal precession rate (for a review see Mazeh, 2008). Unfortunately since BY Dra is not an eclipsing system and our RVs and s are not sufficiently accurate, a measurement of the apsidal motion cannot be carried out. We attempted to fit for but obtained statistically insignificant value.

### 6.3 Rotation pseudo-synchronization

In the case of eccentric orbits one can find a rotational period for which an equilibrium is achieved. This equilibrium, called the pseudo-synchronization, occurs when the ratio of the orbital to the rotational period is:

(5) |

(Hut, 1981; Mazeh, 2008). For the observed eccentricity of BY Dra we get or d, which is in a good agreement with the observed d, and . One may thus conclude that BY Dra is in a rotational equilibrium. The predicted time scale of the pseudo-synchronization is in the case of BY Dra similar but slightly shorter than for the spin-orbit alignment (Hut, 1981). Using the approximate formula for the synchronization time scale of late-type stars given by Devor et al. (2008), we get the value of the order of 10 Myr, so still smaller than the age of the system. The above equation was however derived for binaries with no additional companions (see below) and the tidal evolution of BY Dra AB might be different if the gravitational influence of the third body is taken into account.

### 6.4 Eccentricity and multiplicity of BY Dra

The eccentricity of BY Dra AB () appears to be unusually high. According to Zahn & Bouchet (1989) a circularization of the orbit of a late type system such as BY Dra should occur during the pre-main-sequence phase. This was one of the arguments for the PMS nature of BY Dra. Based on Zahn & Bouchet (1989) we can estimate that in the case of BY Dra the eccentricity should drop to a few percent over years.

However, BY Dra is a hierarchical triple system, with a distant common proper motion companion. The projected separation of 16.7 arc-sec and our distance determination indicate a projected physical separation of 277 AU. As estimated by Zuckerman et al. (1997) its mass is about 0.13 M and assuming a circular orbit, it corresponds to an orbital period of about 2050 years. It is conceivable that the observed eccentricity of the BY Dra AB pair could be explained by the presence of the companion through the Mazeh-Shaham mechanism which results in a cyclic eccentricity variation, known as the Kozai cycles (Kozai, 1962; Mazeh & Shaham, 1979; Fabrycky & Tremaine, 2007; Mazeh, 2008).

Let us denote all the orbital and physical parameters of an unknown perturber by the index . In order to put some constrains on the properties of the perturber which would induce sufficiently strong Kozai cycles, we followed the analysis of Fabrycky & Tremaine (2007). The two main conditions which have to be met in order to produce the observed eccentricity are: (1) a sufficiently large relative inclination of the binary (inner) and the perturber’s (outer) orbit; (2) the Kozai cycles time-scale must be shorter than the period of the inner orbit’s pericenter precession. For the BY Dra AB pair the relativistic precession is the dominant one, being at least an order of magnitude faster than any other (tidal or rotational; Fabrycky & Tremaine, 2007). The precession period is yr, which corresponds to rad s. Using the formalism of Fabrycky & Tremaine (2007) we can estimate that the observed eccentricity of BY Dra can be induced when (in SI units), where the term is computed for . From this we can derive the following conditions for the parameters of the perturbing body:

(6) |

or

(7) |

where the orbital period is given in years, major semi-axis in AU and all masses in M. The condition also allows us to deduce that the relative inclination of the two orbits must be larger than (or smaller than ). This value can be confirmed by the results of Ford, Kozinsky & Rasio (2000), which for the mass ratio of BY Dra AB predict . The relatively narrow range of allows us to put some usefull constrains (which include possible values of the longitudes of ascending nodes) on the absolute value of the perturber’s orbital inclination: .

In Figure 6 we present results of our analysis. We use relation 6 to check whether a body of a given orbital properties (i.e. period and eccentricity) can produce sufficient Kozai cycles. For the mass we have taken mass of a body that in an orbit of a given , , and or 90 would produce an RV modulation of the inner pair at the level of the smaller (157 m s) or half of that. The four panels show the parameter space for the two values of inclinations and RV semi-amplitudes. The shaded areas correspond to the values of and for which the observed eccentricity of BY Dra AB would not be induced. The solid line shows the short-period stability border, calculated in such way that for a given eccentricity the orbit has its periastron at 0.24 AU which refers to the smallest stable circular orbit (with d; Holman & Wiegert, 1999). The long-period cut-off at 2900 days is mainly for the clarity, but it is close to double time span of our observations (1513 d) which means that RV modulations with periods around 3000 d could in principle be detected. The corresponding fourth body detection limits in terms of as a function of its orbital period and eccentricity are presented in the Figure 7.

From those two figures one can deduce that if the observed eccentricity is an effect of the Mazeh-Shaham mechanism, the perturber should be in an orbit of up to single years (major semi axes from to AU) and have its mass in the planetary regime. This is consistent with the fact that the interferometric measurements are fully-consistent with a two-disc model, so no additional light is detected. It is not however excluded that the eccentricity of the perturber is very large which would allow more massive bodies in long period orbits to induce the Kozai cycles and remain undetectable by the RVs. The component C discovered by Zuckerman et al. (1997) could be the perturbing body but with the relation 7 it seems that it is not very likely. For the estimated mass M the orbital eccentricity would have to be larger than 0.98, assuming a fortunate but improbable case that the star is currently seen exactly at the apocenter, and , thus [AU] . For less fortunate cases the value of would have to be even larger.

The other possibility is the presence of a putative fourth body reported in the Hipparcos catalog. Boden & Lane (2001) inspected the available RV data in order to find a 114 day period predicted by the Hipparcos catalog and with a high level of confidence they excluded the existence of such a period in the spectroscopic data. With the relation 6 we can show that if a body in a d circular orbit exists, it would have to have M. From the Figure 7 we can put an upper mass limit of 5.44 M, taking into account the reported inclination of 113.21, which itself is within the allowed limits. The reported major semi-axis of the Hipparcos photocentric orbital solution is AU (at the distance to BY Dra). Assuming the maximum mass ratio , the 4-th body barycentric major semi-axis would be 16.07 AU (after correcting for the inclination), but the Kepler’s 3’rd law predicts the major semi-axis of 0.53 AU for the given period and masses. Such a body would produce the RV signal much stronger than 157 m s. We can thus conclude that the Hipparcos solution is unrealistic.

## 7 Summary

We present the most precise orbital and physical parameters of an important astrophysical object – the low-mass SB2 BY Draconis, a prototype of an entire class of variable stars. We reach a level of precision which allows us to put important constrains on the nature of this object. We conclude that this is not a pre-main-sequence system, despite its high orbital eccentricity and a possible spin-orbit misalignment. However, the gravitational influence of a 4-th, yet undetected body in the system may explain the observed value of and the spin-orbit alignment may be inferred from the available data if a smaller than claimed in the literature value of the rotational velocity is used. The observed rotational period and the eccentricity suggest that the BY Dra AB system is in the rotational equilibrium. However, if the observed eccentricity is indeed due to the presence of the 4-th body, the putatiuve companion may have its mass in the planetary regime. The whole dynamical and tidal picture of this system is more complicated than we previoulsy thought and deserving perhaps a dedicated theoretical, observational and numerical analysis.

## Acknowledgments

We would like to thank Prof. Tsevi Mazeh for his invaluable comments and suggestions, and Arne Rau for carrying out the Keck I/HIRES observations in the years 2006-2007. The authors wish to recognize and acknowledge the very significant cultural role and reverence that the summit of Mauna Kea has always had within the indigenous Hawaiian community. We are most fortunate to have the opportunity to conduct observations from this mountain. This work benefits from the efforts of the PTI collaboration members who have each contributed to the development of an extremely reliable observational instrument. We thank PTI’s night assistant Kevin Rykoski for his efforts to maintain PTI in excellent condition and operating PTI.

This research was co-financed by the European Social Fund and the national budget of the Republic of Poland within the framework of the Integrated Regional Operational Programme, Measure 2.6. Regional innovation strategies and transfer of knowledge - an individual project of the Kuyavian-Pomeranian Voivodship “Scholarships for Ph.D. students 2008/2009 - IROP”, and by the grant N N203 379936 from the Ministry of Science and Higher Education. Support for K.G.H. is provided by Centro de Astrofísica FONDAP Proyecto 15010003, M.K. is supported by the Foundation for Polish Science through a FOCUS grant and fellowship and by the Polish Ministry of Science and Higher Education through grants N N203 005 32/0449 and N N203302035. M.W.M. acknowledges support from the Townes Fellowship Program, an internal UC Berkeley SSL grant, and the State of Tennessee Center of Excellence program. This research was supported in part by the National Science Foundation under Grant No PHY05-51164. The observations on the TNG/SARG have been funded by the Optical Infrared Coordination network (OPTICON), a major international collaboration supported by the Research Infrastructures Programme of the European Commissions Sixth Framework Programme.

This research has made use of the Simbad database, operated at CDS, Strasbourg, France, and of data products from the Two Micron All Sky Survey, which is a joint project of the University of Massachusetts and the Infrared Processing and Analysis Center/California Institute of Technology, funded by the National Aeronautics and Space Administration and the National Science Foundation.

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